In this section, we present the derivation of bounds, summarized in Table REF .
Firstly, to obtain Theorem 2 of [1]}, in (REF ),
we set \(\theta _{\textup {tsk}}=\theta _{\textup {env}}=1\) , and
\(\mathrm {F}^{\textup {Task}} \left(a,b\right)=2(\mathrm {M}-1)(a-b)^2\) , \(\mathrm {F}^{\textup {Env}} \left(a,b\right)=2(-1)(a-b)^2\) . These choices lead to \(k_e=k_t=1\) , \(\mathrm {G}_{\textup {Task}} \left(c\right)=\sqrt{\frac{c}{2(\mathrm {M}-1)}}\)  and also \(\mathrm {G}_{\textup {Env}} \left(c\right)=\sqrt{\frac{c}{2(-1)}}\) . To simplify
\( \textup {B}_\textup {Task}\)  and
\( \textup {B}_\textup {Env}\)  given by (REF ) and (REF ), we use Lemma REF . Since the prior in independent of the data, by interchanging the order of expectations over \(\mathrm {P}_{T_{1:}}\mathrm {P}_{ \mathbf {Z}_{1:}^{\mathrm {M}}|T_{1:}}\)  and priors, in view of (REF ) and (REF ), we find that
\(& \mathbb {E}_{ \mathcal {P}\mathrm {P}} \mathbb {E}_{\mathrm {P}_{T_{1:}}\mathrm {P}_{ \mathbf {Z}_{1:}^{\mathrm {M}}|T_{1:}}}\left[ e^{2(\mathrm {M}-1)\left(\mathrm {L}_{\mathrm {P}_{Z|t_i}}(W)- \mathrm {L}_{\mathbf {Z}_i^{\mathrm {M}}}(W)\right)^2}\right] \le \mathrm {M},\\& \mathbb {E}_{\mathcal {P}} \mathbb {E}_{\mathrm {P}_{ \mathbf {Z}_{1:}^{\mathrm {M}}}}\left[ e^{2(-1)\left(\mathrm {L}_{\mathrm {P}_{T\mathbf {Z}^\mathrm {M}}}(U)-\frac{1}{}\sum _{i=1}^{} \tilde{\mathrm {L}}^{t_i}_{\mathbf {Z}^{\mathrm {M}}_i}(U)\right)^2}\right] \le ,\) 
